direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D5⋊M4(2), D10⋊11M4(2), Dic5.16C24, C5⋊C8⋊2C23, D5⋊(C2×M4(2)), D5⋊C8⋊9C22, C2.5(C23×F5), C10⋊2(C2×M4(2)), C10.3(C23×C4), C4.F5⋊13C22, C5⋊2(C22×M4(2)), (C22×C4).25F5, C23.53(C2×F5), C4.58(C22×F5), (C22×C20).29C4, C20.98(C22×C4), (C23×D5).19C4, (C4×D5).91C23, C22.F5⋊7C22, D10.45(C22×C4), C22.19(C22×F5), Dic5.45(C22×C4), (C2×Dic5).363C23, (C22×Dic5).283C22, (C2×C4×D5).40C4, (C2×D5⋊C8)⋊13C2, (C2×C5⋊C8)⋊10C22, (C2×C4.F5)⋊14C2, (C4×D5).97(C2×C4), (C2×C4).173(C2×F5), (D5×C22×C4).32C2, (C2×C20).152(C2×C4), (C2×C4×D5).405C22, (C2×C22.F5)⋊12C2, (C22×C10).79(C2×C4), (C2×C10).97(C22×C4), (C2×Dic5).199(C2×C4), (C22×D5).133(C2×C4), SmallGroup(320,1589)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D5⋊M4(2)
G = < a,b,c,d,e | a2=b5=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=b3, be=eb, dcd-1=b2c, ce=ec, ede=d5 >
Subgroups: 906 in 298 conjugacy classes, 148 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×4], C22, C22 [×2], C22 [×20], C5, C8 [×8], C2×C4 [×2], C2×C4 [×4], C2×C4 [×22], C23, C23 [×10], D5 [×4], D5 [×2], C10, C10 [×2], C10 [×2], C2×C8 [×12], M4(2) [×16], C22×C4, C22×C4 [×13], C24, Dic5 [×2], Dic5 [×2], C20 [×4], D10 [×8], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C22×C8 [×2], C2×M4(2) [×12], C23×C4, C5⋊C8 [×8], C4×D5 [×16], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, C22×M4(2), D5⋊C8 [×8], C4.F5 [×8], C2×C5⋊C8 [×4], C22.F5 [×8], C2×C4×D5 [×4], C2×C4×D5 [×8], C22×Dic5, C22×C20, C23×D5, C2×D5⋊C8 [×2], C2×C4.F5 [×2], D5⋊M4(2) [×8], C2×C22.F5 [×2], D5×C22×C4, C2×D5⋊M4(2)
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×4], C22×C4 [×14], C24, F5, C2×M4(2) [×6], C23×C4, C2×F5 [×7], C22×M4(2), C22×F5 [×7], D5⋊M4(2) [×2], C23×F5, C2×D5⋊M4(2)
►On 80 points
(1 79)(2 80)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(17 69)(18 70)(19 71)(20 72)(21 65)(22 66)(23 67)(24 68)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 33)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 57)(56 58)
(1 58 70 40 12)(2 33 59 13 71)(3 14 34 72 60)(4 65 15 61 35)(5 62 66 36 16)(6 37 63 9 67)(7 10 38 68 64)(8 69 11 57 39)(17 44 55 30 78)(18 31 45 79 56)(19 80 32 49 46)(20 50 73 47 25)(21 48 51 26 74)(22 27 41 75 52)(23 76 28 53 42)(24 54 77 43 29)
(1 16)(2 67)(3 64)(4 39)(5 12)(6 71)(7 60)(8 35)(9 33)(10 72)(11 15)(13 37)(14 68)(17 51)(18 22)(19 76)(20 43)(21 55)(23 80)(24 47)(25 29)(26 78)(27 56)(28 46)(30 74)(31 52)(32 42)(34 38)(36 58)(40 62)(41 79)(44 48)(45 75)(49 53)(50 77)(54 73)(57 65)(59 63)(61 69)(66 70)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)
(1 75)(2 80)(3 77)(4 74)(5 79)(6 76)(7 73)(8 78)(9 42)(10 47)(11 44)(12 41)(13 46)(14 43)(15 48)(16 45)(17 69)(18 66)(19 71)(20 68)(21 65)(22 70)(23 67)(24 72)(25 38)(26 35)(27 40)(28 37)(29 34)(30 39)(31 36)(32 33)(49 59)(50 64)(51 61)(52 58)(53 63)(54 60)(55 57)(56 62)
G:=sub<Sym(80)| (1,79)(2,80)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(17,69)(18,70)(19,71)(20,72)(21,65)(22,66)(23,67)(24,68)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,33)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58), (1,58,70,40,12)(2,33,59,13,71)(3,14,34,72,60)(4,65,15,61,35)(5,62,66,36,16)(6,37,63,9,67)(7,10,38,68,64)(8,69,11,57,39)(17,44,55,30,78)(18,31,45,79,56)(19,80,32,49,46)(20,50,73,47,25)(21,48,51,26,74)(22,27,41,75,52)(23,76,28,53,42)(24,54,77,43,29), (1,16)(2,67)(3,64)(4,39)(5,12)(6,71)(7,60)(8,35)(9,33)(10,72)(11,15)(13,37)(14,68)(17,51)(18,22)(19,76)(20,43)(21,55)(23,80)(24,47)(25,29)(26,78)(27,56)(28,46)(30,74)(31,52)(32,42)(34,38)(36,58)(40,62)(41,79)(44,48)(45,75)(49,53)(50,77)(54,73)(57,65)(59,63)(61,69)(66,70), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,75)(2,80)(3,77)(4,74)(5,79)(6,76)(7,73)(8,78)(9,42)(10,47)(11,44)(12,41)(13,46)(14,43)(15,48)(16,45)(17,69)(18,66)(19,71)(20,68)(21,65)(22,70)(23,67)(24,72)(25,38)(26,35)(27,40)(28,37)(29,34)(30,39)(31,36)(32,33)(49,59)(50,64)(51,61)(52,58)(53,63)(54,60)(55,57)(56,62)>;
G:=Group( (1,79)(2,80)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(17,69)(18,70)(19,71)(20,72)(21,65)(22,66)(23,67)(24,68)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,33)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58), (1,58,70,40,12)(2,33,59,13,71)(3,14,34,72,60)(4,65,15,61,35)(5,62,66,36,16)(6,37,63,9,67)(7,10,38,68,64)(8,69,11,57,39)(17,44,55,30,78)(18,31,45,79,56)(19,80,32,49,46)(20,50,73,47,25)(21,48,51,26,74)(22,27,41,75,52)(23,76,28,53,42)(24,54,77,43,29), (1,16)(2,67)(3,64)(4,39)(5,12)(6,71)(7,60)(8,35)(9,33)(10,72)(11,15)(13,37)(14,68)(17,51)(18,22)(19,76)(20,43)(21,55)(23,80)(24,47)(25,29)(26,78)(27,56)(28,46)(30,74)(31,52)(32,42)(34,38)(36,58)(40,62)(41,79)(44,48)(45,75)(49,53)(50,77)(54,73)(57,65)(59,63)(61,69)(66,70), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,75)(2,80)(3,77)(4,74)(5,79)(6,76)(7,73)(8,78)(9,42)(10,47)(11,44)(12,41)(13,46)(14,43)(15,48)(16,45)(17,69)(18,66)(19,71)(20,68)(21,65)(22,70)(23,67)(24,72)(25,38)(26,35)(27,40)(28,37)(29,34)(30,39)(31,36)(32,33)(49,59)(50,64)(51,61)(52,58)(53,63)(54,60)(55,57)(56,62) );
G=PermutationGroup([(1,79),(2,80),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(17,69),(18,70),(19,71),(20,72),(21,65),(22,66),(23,67),(24,68),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39),(31,40),(32,33),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,57),(56,58)], [(1,58,70,40,12),(2,33,59,13,71),(3,14,34,72,60),(4,65,15,61,35),(5,62,66,36,16),(6,37,63,9,67),(7,10,38,68,64),(8,69,11,57,39),(17,44,55,30,78),(18,31,45,79,56),(19,80,32,49,46),(20,50,73,47,25),(21,48,51,26,74),(22,27,41,75,52),(23,76,28,53,42),(24,54,77,43,29)], [(1,16),(2,67),(3,64),(4,39),(5,12),(6,71),(7,60),(8,35),(9,33),(10,72),(11,15),(13,37),(14,68),(17,51),(18,22),(19,76),(20,43),(21,55),(23,80),(24,47),(25,29),(26,78),(27,56),(28,46),(30,74),(31,52),(32,42),(34,38),(36,58),(40,62),(41,79),(44,48),(45,75),(49,53),(50,77),(54,73),(57,65),(59,63),(61,69),(66,70)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80)], [(1,75),(2,80),(3,77),(4,74),(5,79),(6,76),(7,73),(8,78),(9,42),(10,47),(11,44),(12,41),(13,46),(14,43),(15,48),(16,45),(17,69),(18,66),(19,71),(20,68),(21,65),(22,70),(23,67),(24,72),(25,38),(26,35),(27,40),(28,37),(29,34),(30,39),(31,36),(32,33),(49,59),(50,64),(51,61),(52,58),(53,63),(54,60),(55,57),(56,62)])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5 | 8A | ··· | 8P | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) | F5 | C2×F5 | C2×F5 | D5⋊M4(2) |
| kernel | C2×D5⋊M4(2) | C2×D5⋊C8 | C2×C4.F5 | D5⋊M4(2) | C2×C22.F5 | D5×C22×C4 | C2×C4×D5 | C22×C20 | C23×D5 | D10 | C22×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 8 | 2 | 1 | 12 | 2 | 2 | 8 | 1 | 6 | 1 | 8 |
Matrix representation of C2×D5⋊M4(2) ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 35 |
| 0 | 0 | 0 | 0 | 6 | 35 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 1 |
| 0 | 0 | 0 | 0 | 6 | 35 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 32 | 28 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,6,0,0,0,0,0,0,40,6,0,0,0,0,35,35],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,1,0,0,0,0,6,6,0,0,0,0,0,0,6,6,0,0,0,0,1,35],[40,4,0,0,0,0,2,1,0,0,0,0,0,0,0,0,32,0,0,0,0,0,28,9,0,0,40,0,0,0,0,0,0,40,0,0],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C2×D5⋊M4(2) in GAP, Magma, Sage, TeX
C_2\times D_5\rtimes M_4(2)
% in TeX
G:=Group("C2xD5:M4(2)"); // GroupNames label
G:=SmallGroup(320,1589);
// by ID
G=gap.SmallGroup(320,1589);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,1123,102,6278,818]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=b^3,b*e=e*b,d*c*d^-1=b^2*c,c*e=e*c,e*d*e=d^5>;
// generators/relations